A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Saturday, March 29, 2008

I returned just a few hours ago from the MAA Spring Southeast Sectional Meeting held this year at The Citadel in Charleston, SC. (Two conferences in Charleston, in one year!) I spent much of my time with four of our stalwart students, three of whom presented posters in this morning's undergraduate poster session...

...I'll have much more to say about the conference, likely tomorrow...but for now, I'm fair and squarely exhausted and so must bid adieu...

Wednesday, March 19, 2008

It's Wednesday. It's been raining all day, and the rain has hardly ceased, even now that the sun is down.

We're one student away from filling our last slot in this upcoming summer's REU, having received a seventh acceptance today. I spent an hour or so this afternoon hammering out a list of learning goals for the program, and a schedule of activities through which we will work towards realizing those goals. Like the goals I typically set for my classes, the list includes content-oriented targets like mastery of graph theory, group theory, etc., but also less traditional goals such as gaining confidence in communicating mathematics to others, and building the authority to challenge unproven results.

How well will we fare?

Well, how'd we do last year? We can't possibly do worse, can we? (Famous last words...) Here's a brief report card on 2007, filled out from the perspective offered by a year of hindsight:

Recruitment: A. We got great kids, and what's so marvelous is that we did so so late in the game, having not secured funding until a month or so after most REUs had already made their hires. That delay allowed us to catch the best of the best of the younger crowd, many of whom had missed the first round of REU applications. We lucked out. I feel honored that I had the chance to work with such a talented group of students, many of whom are surely destined for great things.

Logistics: A. We handled housing well, we covered all the human resource aspects admirably. As far as I'm aware (aside from one snafu with one of the subsistence checks when something didn't get signed in time), all of the paperwork came off without a hitch. Yay, we're good pencil-pushers!

Seminar: B+. For the most part, we hit the nail on the head. I think the structure of our opening week-and-a-half was sound, and it did a good job of preparing the students for what would come in the next weeks. I don't think we adequately anticipated the stress it would induce in some (all?) of the participants...but we adjusted for that, and pulled up in time. This time around we'll know what to expect, we'll be able to ease up when needed.

Structure: B+. Again, things moved along smoothly, for the most part. The students did a great job of establishing semi-regular meeting times with their respective faculty mentors, the students did a great job in keeping their noses to the grindstones, the weekly meetings were generally productive. Those meetings, though, were awkwardly scheduled, and I'm not sure that the students played as strong a leading role as they could have: in the future, we might be able to challenge the students to take authority in these sessions. Moreover, we didn't have a chance to include any "guest" research talks by faculty from UNCA or elsewhere, as we'd hoped we'd be able to do. (This I've already remedied this time around: I've sent out three invitations to colleagues from other institutions, and have already received one positive reply.)

Social organization: A. We couldn't have done it without the students, who got along admirably well. Not only did they not kill each other, they became fast friends. By the summer's close, the care and concern they showed for one another was evident. (I hope this year's crowd will come to the conclusion that they need to have a talent show, too...although nothing's going to top the 2007 crew's rendition of "A Whole New World.")

Research outcome: Incomplete. It's hard to say at this point how "much" the students will have generated when all the dust has settled. Wilhelmina and Francoise have got a nearly submission-ready manuscript they've been sitting on for a while now, Ned's work with me will make a nice section in the paper whose prequel has been tentatively picked up by a nice graph theory journal, and Kendrick's name appears on an as yet unsubmitted manuscript my next-door colleague here has put together. Let's hold off on this one.

Long-term outcome: Another incomplete. I'm not trying to cop out here; we're still too close to this past summer to measure long-term outcomes, but I like to think that the program made a primarily positive impact on the budding careers of a handful of talented young mathematicians. As far as I'm concerned, if five years from now I run into one of the participants at a conference just after she's presented on her dissertation research, and she's able to say that her experience was a worthwhile one and helped her decide what she wanted to do with her career, then we've dealt ourselves a royal flush.

Changes this year? Reflecting changes in my own pedagogical style over the past year or two, the program already exhibits more conscious design and attention to explicitly stated learning goals. Writing plays a more central role, with an introduction to LaTeX coming in at the program's beginning (towards the end of the first week) rather than at its end (towards the end of the sixth week). Indeed, written progress reports will be expected of this year's students, in addition to the weekly meetings. We're also going to make meeting times explicit, and as I mentioned above we've already begun scheduling guest speakers. Finally, and perhaps most importantly, we'll be encouraging the students to seek out their own problems more actively: though we'll still have ready stockpiles of personal problems from which the students will be able to draw, the participants will be encouraged to seek out problems that entice them, hopefully from within the fields in which the participating faculty specialize. "All right, y'all, that's everything you need to know about chromaticity of Cayley graphs. Here's a survey paper. Dig in!"

I'm excited. Now that we're in the thick of it instead on the fringe, this year's selection process has been more of a roller coaster than last year, and with as many noes as yeses the thrill of the chase has gotten my blood pumping. We'll have our team set up soon, and I'll probably take one more shot at getting folks to blog about themselves before they get here. (Last year's awful attempt failed pitifully...I'm pretty sure that I mercifully deleted the pathetic little webpage that limped along painfully for a few weeks...yup! Just checked: all gone.)

What else is up?

For a few weeks now I've meant to say a bit more about my Graph Theory class, let me take this time to do so.

In-class presentations are for the most part much improved, especially in the past few weeks. The students who take the time to craft solid proofs ahead of time are executing marvelous performances and are sometimes uncovering techniques I would not have considered. Though their methods are not always the most efficient, they're authentic, through and through. Today in particular saw a handful of nearly immaculate proofs: one problem asked the students to prove that the path metric induced by a subgraph could only exceed the path metric of the original supergraph, another asked for an explanation for what breaks down when one tries to define the path metric on a disconnected graph, yet others asked properties of eccentricity. All solutions were skilfully executed.

Where some of the students are having trouble is in the written submissions. How so? Well, c'mon, people, even if all the problem says is "find the chromatic polynomial of the complete graph on n vertices," I can't jolly well in good conscience give the same grade to some who just hands me the formula, ex nihilo, as to someone who includes a half-page proof of that same formula. Trust me, from now on I'm going to explicitly include wording like "give a formula for... and prove that your formula is valid." I'll say that, if you'd like me to, but I claim that at this point I shouldn't have to say this, it ought to be assumed that at this level we prove our claims.

But we all know that when we assume, we make an ass out of "u" and me.

For the most part, Graph Theory's a blast. I'm still having fun, I think most of the students are finding it a worthwhile experience and are learning a lot. For Friday, I've asked them each to write a few paragraphs about what they feel is working well, and what could stand to be changed for the closing third or so of the semester. I'm eager to see what they've got to say. I've already talked to two of them about modifying the "review" problems at the end of each problem sheet, to allow these problems to be more group-centered and in-class. We'll see we can make that work, if others are up for trying it out.

It's all good.

I'm getting tired, I'm going to slink off in a moment, but just a quick word about my Calc II kiddies: two days into sequences, they're doing great. "These are fun!" one of my students said. They've already asked great questions and have exhibited profound intuition and insight. I think they'll be able to wrap their minds around this stuff comfortably. I'm also happy to report that Taylor series are playing a crucial role in my own ongoing research right now, so I should be able to bring that in as a "real-world" example of series methods before the semester's through. Huzzah! This crap is useful! Who'd'a thunk it?

Well, more anon, likely. For now, I'm off like a prom dress, as my college buddy Jennifer was fond of saying. Ta for now.

Sunday, March 16, 2008

I've now had two days to recover from the hedonistic revelry that accompanied this year's observance of that most hallowed of days, Pi Day, March 14th, and I've a few minutes of time in which to sit down and chronicle the occasion.

This is the second straight year we've put a bit of effort (more this year than last) into celebrating this immovable mathematical feast, and that effort paid off, with roughly fifty folks, mostly Math Department students and faculty and their close acquaintances, in attendance at the 1:59 ceremonies.

What went on?

For some weeks now Stanley (our Math Club president) and I have been mulling over various means of approximating π probabilistically that would lend themselves to audience participation. The classic Buffon's needle experiment (implemented here on George Reese's homepage at the University of Illinois, Urbana-Champaign) could be replicated by allowing passersby to chuck hot dogs into an enclosure with a ruled surface, enabling a running tally of hot dogs that strike a line. The cost of implementing this procedure would be rather high, unless we wanted to reuse the same hot dogs over and over and over(an icky proposition)...plus there's the need for constant supervision of the enclosure, and we've called upon our students quite a bit lately, what with the recent Math Literacy Summit. To bring the cost down, we thought then about replacing the hot dogs with pixie sticks, which would be more inexpensive and likely more accurate (there would be less error incurred by the thinner width of the pixie sticks), but we'd still have to ensure the event was continually monitored, in order to tally up the results of the experiment.

Then I hit upon the idea of just doing a simple Monte Carlo area estimate: build a small square enclosure, and let people chuck spare change into it throughout the day. At the day's end, collect all of the coins that lay within a circle centered at the enclosure's middle, and divide by the total number of coins present. This ratio should be roughly π/4, the ratio of the circle's area to that of the square. Assuming a fair degree of faith in human nature, there'd be no need to oversee the experiment, since the coins would only minimally interfere with one another. All we'd have to do is put the booth up in the morning and take it down at night after carefully cataloging the location of the coins.

This we did. I spent a few minutes on the evening of the 13th drilling holes in the plywood and posts, and then Maggie and I schlepped the assemblage up to campus, along with the roughly 18 pounds of pie we'd bought for the pie-eating to take place on the following afternoon.

First thing on Friday morning, I went downstairs and slapped the enclosure together. The edges bowed outward slightly, but it was very roughly square and would serve well. I tacked an explanatory note to each side of the enclosure, along with an encouragement for people to chuck their change into the square:

Classwise, it was a humdrum day. For whatever reason (I attributed it to hangovers resulting from an overly exuberant demarcation of Pi Eve the night before) attendance at my morning Calc II class was exceptionally bad, and I felt no qualms in devoting the class period to working on the current class project, asking students to compute the centroids of various pieces of poster board. (Funny story about that project: as I handed out the project this past Tuesday, Louella asked me, "so, were you a creative writing minor in college?" when she read the project description, with the following text: "...the Math Lab will be home to eight small shapes cut from festively-colored poster board. (They are bundled together with a binder clip, hanging from a tack over by the coffee pots.) There are three colors represented: four are blood red, three are Day-Glo orange, and a final shape is a nice soothing green, as fresh as a newborn magnolia leaf.") The class was a relaxing one, the students were laid back, and had fun working together to get a good head start on the project. I hope Monday's class will be similar, when I'll circulate various worksheets asking the students to consider various applications of integrals not considered by the textbook.

The second section of Calc II was as fun as the first, and we wrapped up just in time for me to bolt upstairs to gather what we'd need to set up for the 1:59 celebration: pies and plates, camera and stopwatch, prizes, and a print-out of π to a thousand places. Even as I came back down from the first of two trips to my office, students and faculty were beginning to gather. By the time the ceremony got underway with a dramatic reading of π (pictured below) there were about thirty or forty people assembled, and more still would come to watch the pie-eating contest in a few more minutes.

After a few words about the occasion, I began the reading of π with a bold recitation of the first 25 places, handing the script to a student who would continue where I'd left off. Emotions bubbled close to the surface as student after student took turns reading digits.

Here's a shot showing the thrilling denouement of the dramatic reading:

Next came the pie-eating. Five stalwart students came forth to vie voraciously, and each was seated with a pound of pie in front of him or her (four hims, one her). At the appointed moment, they set to, chomping away for 3 minutes and 14 seconds.

At the end of the carnage, little was left of most of the pies but the skeletons of empty crusts. Norbert, an engineering student in my first Calc II class, was declared by the several faculty judges to be the winner, with Nicodema, the contest's sole female entrant, coming in a close second. The fearsome five gathered for a group photo at the contest's end:

Next came the π-memorizing contest (more appropriately, perhaps, the π-reciting-from-memory contest). Just that day I'd announced to both of my Calc II classes that if they sat down and committed twenty or thirty places to memory they'd probably stand a good chance of winning the competition, since I expected a fairly weak field. Little did I know that last year's winner, Ulrich, had returned to defend his championship. He intended to best his previous record of 64 places with a public recital of the first 150 places of π.

Loath to let Ulrich get away without a challenge, Trixie came forward and belted out 48 places unerringly, offering an incredible extemporaneous memorization. Here she is, the midst of her performance:

Her recitation was followed by a flawless 45 places, and it was then up to Ulrich to hold his own.

This he did, rattling off 150 places with only the slightest pause now and then. Here he is below, in the midst of his recital:

After this, there was plenty of opportunity for hangers-on to mingle and partake of a leisurely piece of pie themselves. My department chair then gathered everyone present and took a photo of the whole throng. I count 44 people in the picture, and I know there were at least four present who were not captured "on film" (what does one say these day? "On flash" doesn't have quite the same ring to it...):

So it is that with heavy hearts we say farewell to another Pi Day, only to wait another year before again marking this felicitous occasion. (By the way, by the day's end, our enclosure had gathered over 300 coins, yielding an estimate for π that was around 2.79. I've got the data in my office, I'll post them later when they're in front of me.)

After a brief bit of frenzied clean-up, I was off to Graph Theory. There we finally managed to finish off the now notorious Problem Sheet 7, dealing with chromatic polynomials, components, and the basics of trees. Proof-heavy and definition-intensive, this sheet was a definite departure from the previous ones, and it challenged even the strongest students in the class. "Now that we've got the fundamental of graph theory under our belts, we're able to consider some of the deeper concepts and techniques, and that's what this sheet has been asking us to do," I told them. We're now set to begin the next sheet, in which is introduced and investigated the path metric on a given graph. That's where we'll find ourselves on Monday.

I regret that I've not had the time lately to update this blog as much as I'd like to...and when I've had the time, I've hardly had the strength, as busy as I've been. Now that the Numeracy Summit has passed, and now that the bulk of work on our NSF grant is completed, and now that the REU applications have been read and evaluated (we're about halfway through the selection process as I write this), I will likely have a lot more time on my hands, and I'll be less tired when I have it.

I've got a good deal of travel coming up, about which I'm very excited. For instance, in a couple of weeks it'll be down to Charleston for the Southeastern Sectional MAA Meeting, several students in tow and several colleagues by my side. Trixie will be presenting a poster there on her work in graceful graphs, I'm very proud. Whether or not she chooses to pursue a math degree, this experience will be a fantastic one for her.

Aside from travel, there's the REU to get ready for (we're opting for a "less directed" approach this year, offering students a bit more room to explore), and the Parsons Lecture (featuring Mary Lou Zeeman) is just a few weeks away. Much to do, much to do!

Sunday, March 02, 2008

Which doesn't, sadly, mean I have nothing to do. It only means that what I've got to do (and there is a great deal of it) needn't be done on a rigid schedule.

I've got several job-related tasks to take care of in the next week, ranging from the quotidian (prepping for class once school resumes a week from tomorrow) to the leviathan (going through a stack of roughly 75 applications for this coming summer's REU). I've got a couple of meetings tomorrow, one with a student (my independent study in order theory), another with a colleague (Writing Intensive stuff). After that, I'm looking at a nearly completely unstructured week.

I'm in need of some unstructured time, after the busyness of this past week. Dr. Robert P. Moses, noted civil rights leader and founder of the Algebra Project, came for his visit this past Wednesday, and between the public lecture on Wednesday evening (a talk about the degree to which the Constitution ensures a quality education, at which I was delighted to see several of my students!) and the ensuing Math Literacy Summit held on Thursday, there was no shortage of excitement and things to do in our department.

The session I chaired at the summit (a talk on numeracy as it relates to health issues, given by a psychologist at the Duke University Medical Center) led me to the book I'm now reading, Stanislas Dehaene's The number sense: how the mind creates mathematics (Oxford University Press, 1997). This is proving a truly fascinating read!

Dehaene is a psychologist specializing in the neurobiology of mathematical acquisition, his book is a record of many of the facts that have been discovered concerning the way in which people learn mathematics, they way they organize its ideas in our minds, the way math is retrieved from memory. At its most basic level, our sense of mathematics is very little advanced beyond that of many animals, who share with us a precise sense only of the numbers 1, 2, and 3; beyond this is a roughly-reckoned haze of numeric quantities. Dehaene compares our mental conception of number as an "accumulator" with approximate graduations allowing us to give rough estimates of large quantities, but which fails to give precise values for these same quantities.

A few snippets:

Even as soon as a few days after birth, babies are able to discern between the numbers 2 and 3. (See p. 50.)

We (adults included!) are susceptible to "the magnitude effect": it's harder for us to discern the difference between 90 objects and 100 than it is the difference between 10 objects and 20. Various factors (symmetry, density, etc.) militate and mitigate this effect. (See pp. 71 ff.)

Studies show that when asked to compare numbers, such as 5 and 7, and state which is the larger, instead of behaving reflexively and answering based upon our knowledge that the symbol "7" represents a larger quantity than the symbol "5," we instead convert each of these abstract digits into collections of the requisite number of objects before performing the comparison on these collections. (See pp. 75 ff.)

We have a tendency to "compress" numbers as they grow, storing them in our minds as though on a logarithmic scale. One corollary of this behavior is that when asked to provide a random sample of numbers in a certain range, people will tend to elect an overrepresentation of smaller values, as though these were more widely spaced than their larger compatriots. (See pp. 77 ff.)

Since adults compute sums and products (for example) by retrieving the resultant quantity from a memorized table, those whose native languages have exceedingly short names for the ten numerals (like Chinese and Japanese) are able to more efficiently memorize the desired sums and products, and so perform much more quickly and with fewer errors than their counterparts speaking other tongues. (See pp. 130 ff.)

These are just a few of the fascinating facts I'm learning about the development and refinement of mathematical thought processes in and by the human mind. Ultimately, one of Dehaene's primary points is summed up nicely on pp. 118-119: "Although our knowledge of this issue is still far from complete, one thing is certain: Mental arithmetic poses serious problems for the human brain. Nothing ever prepared it for the task of memorizing dozens of intermingled multiplication facts, or of flawlessly executing the ten or fifteen steps of a two-digit subtraction. An innate sense of approximate numerical quantities may well be embedded in our genes; but when faced with exact symbolic calculation, we lack proper resources."

To be continued, I'm sure.

For now, I'm off to enjoy some more of this wonderfully unstructured time, probably by knocking off a few more pages of Dehaene's book. Highly recommended!